Mod p Hecke algebras and dual equivariant cohomology I: the case of $G{L}_2$

Let $F$ be a p-adic local field and $G=G{L}_2(F)$. Let ${\mathcal{H}}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of ${\mathbb{F}}_p$. We show that the supersingular irreducible ${\mathcal{H}}^{(1)}$-modules of dimension 2 can be realized through the equivariant cohomology of the flag variety of the mod p Langlands dual group of $G$.